I think you should just do whatever is easiest. Unless your cat is a purebred and you plan on doing cat shows or breed it, I don't think it matters if its medical records match its name. Even then I don't know anything about cat shows, so maybe not. And don't rule everyone out, your vet might be a hobbit fan, or they might not care at all what you call your cat.

My cat knows his name, but I think he thinks his name is just the word I use to mean "stop whatever you are doing right now" because I usually only say it when he's jumped on the counter or is scratching the stairs. He may think his actual name is "Here-Kitty-Kitty" because that's what I say when I actually want to pet him.

Well, even if education isn't a priority in your education, I hope your vet is educated! Just call him what you want, I'm sure vets are used to weird names (not that Smaug is weird, just sayin' vets probably hear a lot of weird names).